# Number of matrices having unique solution

Gold Member

## Homework Statement

Let A be the set of all 3x3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices B in A for which the system of linear equations $B \left[ \begin{array}{c} x \\ y \\ z \end{array} \right] = \left[ \begin{array}{c} 1 \\ 0 \\0 \end{array} \right]$ has a unique solution is

## The Attempt at a Solution

I started by finding total number of matrices in A which comes out to be 12. Now, instead of finding matrices having unique solution I tried finding out inconsistent solutions. Now, the problem arises that how do I find these matrices. Is it by writing out all possible 12 matrices and then checking manually or applying some clever methods (which I unfortunately don't know)?

pasmith
Homework Helper

## Homework Statement

Let A be the set of all 3x3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices B in A for which the system of linear equations $B \left[ \begin{array}{c} x \\ y \\ z \end{array} \right] = \left[ \begin{array}{c} 1 \\ 0 \\0 \end{array} \right]$ has a unique solution is

This equation has a unique solution if and only if $\det B \neq 0$.

AlephZero
Homework Helper
Find out some facts about determinants. For example, what happens to the determinant if you swap two rows or two columns of the matrix? What happens if two rows or columns of the matrix are the same?

Using results like the above, you won't need to check all 12 matrices separately.

Gold Member
This equation has a unique solution if and only if $\det B \neq 0$.

Why is it so?

haruspex